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Chapter 4: Problem 82

$$\text { Find } \lim _{x \rightarrow \infty}(\sqrt{x^{2}+1}-\sqrt{x})$$.

Short Answer

Expert verified
The limit is 0.

Step by step solution

01

Identify the Problem Type

This is a limit problem where we need to find the behavior of a function as \( x \) approaches infinity.
02

Recognize Dominant Terms

For large values of \( x \), the dominant term in \( \sqrt{x^2+1} \) is \( x \), because it grows faster than 1. This suggests simplifying the expression for large \( x \).
03

Simplify the Expression

To simplify \( \sqrt{x^2+1} - \sqrt{x} \), multiply the expression by the conjugate: \( \frac{\sqrt{x^2+1} - \sqrt{x}}{1} \times \frac{\sqrt{x^2+1} + \sqrt{x}}{\sqrt{x^2+1} + \sqrt{x}} \).
04

Multiply and Simplify

The product of \( (\sqrt{x^2+1} - \sqrt{x})(\sqrt{x^2+1} + \sqrt{x}) \) simplifies to \( (x^2+1) - x = x(x^2+1) - x = 1 \).
05

Evaluate Limit

The expression becomes \( \frac{1}{\sqrt{x^2+1} + \sqrt{x}} \). As \( x \rightarrow \infty \), \( \sqrt{x^2+1} \sim x \) and \( \sqrt{x} \sim \sqrt{x} \), so\( \sqrt{x^2+1} + \sqrt{x} \rightarrow x + \frac{x}{2} = \frac{3x}{2} \).
06

Calculate the Limit

Thus, the expression \( \frac{1}{\sqrt{x^2+1} + \sqrt{x}} \approx \frac{1}{x + \frac{x}{2}} \to 0 \) as \( x \to \infty \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinity
In mathematics, the concept of infinity often appears when studying the behavior of functions as the input values grow extremely large or small. In our problem, we're interested in the limit of an expression as \( x \) approaches infinity. This process helps to understand how a function behaves at extreme values.
  • As \( x \) gets very large, certain terms in an expression become negligible compared to others.
  • We focus on those terms that influence the expression the most, known as dominant terms.
By exploring the expression as \( x \) trends toward infinity, we gain insights into its limiting behavior, allowing us to simplify and determine its value or behavior in a more approachable manner.
Dominant Terms
When evaluating limits as \( x \) tends toward infinity, identifying the dominant terms in an expression is key. Dominant terms are those parts of the expression that increase or decrease the fastest and have the most significant impact on the expression's value.
  • Consider the expression \( \sqrt{x^2+1} \). As \( x \) increases, \( x^2 \) becomes much larger than 1, so \( \sqrt{x^2+1} \) behaves similar to \( x \).
  • This simplification strategy makes it easier to predict the behavior of the overall expression as \( x \) grows large.
Identifying and focusing on these dominant terms allows us to accurately simplify complex expressions and determine their limits at infinity.
Conjugate Multiplication
Conjugate multiplication is a technique used to simplify expressions, especially when dealing with square roots or radicals. In this method, we multiply the expression by a form of 1鈥攊n this case, the conjugate of the expression.
  • For the expression \( \sqrt{x^2+1} - \sqrt{x} \), we use its conjugate \( \sqrt{x^2+1} + \sqrt{x} \).
  • The multiplication of these conjugates results in a difference of squares: \( (\sqrt{x^2+1})^2 - (\sqrt{x})^2 = x^2+1 - x \).
This method simplifies the expression by removing the radicals, making it easier to manage.
Expression Simplification
Simplifying expressions often involves eliminating complexity, particularly with large value limits. After employing conjugate multiplication, we strive to simplify the expression further. With our expression:
  • After multiplying by the conjugate, we reduce the expression to \( \frac{1}{\sqrt{x^2+1} + \sqrt{x}} \).
  • As \( x \to \infty \), recognizing that \( \sqrt{x^2+1} \sim x \), the expression simplifies to \( \sim \frac{1}{x+\frac{x}{2}} = \frac{1}{\frac{3x}{2}} \).
This results in a straightforward limit calculation, showing that as \( x \to \infty \), the expression approaches zero. Simplifying expressions in this way helps visualize and determine limits efficiently.

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Most popular questions from this chapter

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=\pi x^{2} e^{-3 x / 2}, \quad[0,5]$$

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Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int x \sin x \, d x=\frac{x^{2}}{2} \sin x+C\) b. \(\int x \sin x \, d x=-x \cos x+C\) c. \(\int x \sin x \, d x=-x \cos x+\sin x+C\)

The standard equation for the position \(s\) of a body moving with a constant acceleration \(a\) along a coordinate line is $$ s=\frac{a}{2} t^{2}+v_{0} t+s_{0} $$ where \(v_{0}\) and \(s_{0}\) are the body's velocity and position at time \(t=0 .\) Derive this equation by solving the initial value problem Differential equation: \(\frac{d^{2} s}{d t^{2}}=a\) Initial conditions: \(\frac{d s}{d t}=v_{0}\) and \(s=s_{0}\) when \(t=0\)
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